In this paper, we present an efficient rescaling scheme for computing the long-time dynamics of expanding interfaces. The idea is to design an adaptive time-space mapping such that in the new time scale, the interfaces evolves logarithmically fast at early growth stage and exponentially fast at later times. The new spatial scale guarantees the conservation of the area/volume enclosed by the interface. Compared with the original rescaling method in [J. Comput. Phys. 225(1) (2007) 554–567], this adaptive scheme dramatically improves the slow evolution at early times when the size of the interface is small. Our results show that the original three-week computation in [J. Comput. Phys. 225(1) (2007) 554–567] can be reproduced in about one day using the adaptive scheme. We then present the largest and most complicated Hele-Shaw simulation up to date.

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$ , given $g$ . There are several approaches involving the use of series expansions of derivatives of $g$ , which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$ ) for the convergence of the series concerned.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

The integral representation for the solution of the 2-D Dirichlet problem for harmonic functions with boundary data on closed and open curves is obtained. The solution is expressed as a sum of potentials, the density of which obeys the uniquely solvable Fredholm integral equation of the second kind.

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

The authors begin by presenting a brief survey of the various useful methods of solving certain integral equations of Fredholm type. In particular, they apply the reduction techniques with a view to inverting a class of generalized hypergeometric integral transforms. This is observed to lead to an interesting generalization of the work of E. R. Love [9]. The Mellin transform technique for solving a general Fredholm type integral equation with the familiar H-function in the kernel is also considered.

For Fredholm equations of the first kind with continuous kernels we investigate the uniform convergence of a general class of regularization methods. Applications are made to Tikhonov regularization and Landweber's iteration method.